few numerical methods question ... ?

[rosekidcute]

what does these two mean ?

??

 

[stapel]

What do you mean by "what do they mean"? Which part, in particular, is causing you confusion? Please be very specific! Thank you!

 

[rosekidcute]

i am new to differential equations and numerical methods ...

the question and the answer part is a bit confusing ..

if the question is to find the function that gives , this instantaneous rate of change ...

the answer is the function that has the property of this rate of change ... right ??

what does that third pictures answer really mean ...

that part i cant put it in plain english ...


please help ...

 

[stapel]

i am new to differential equations and numerical methods ...

the question and the answer part is a bit confusing ...

...i cant put it in plain english ...

Um... If you don't understand the material and don't know how to state your question, I'm afraid there's really little way for anybody here to attempt an answer. You may want to consider hiring a qualified local tutor who can provide the person lesson instruction that you seem to be seeking.

Good luck!

 

[rosekidcute]

most people i would ask wouldn't know much about these ...

my college ended in 2009 too ...

is this some sort of algorithm ??

 

[rosekidcute]

i have few more questions ...

we had this "computer oriented numerical methods in c language "

it was a difficult subject to follow ...

i was getting stuck without proper materials ...

we had a messed up syllabus ...

so i have been trying to re arrange the whole syllabus , so that it becomes something nice to look at and easy to learn ...

it was supposed to start with a program for a polynomial factorization first ... but i cannot even find one example program of a polynomial factorization in c in any texts or online...

i have been following at least this much amount of books to narrow it down ...

Peter Selby, Steve Slavin Practical Algebra_ A Self-Teaching Guide
Tom M. Apostol Calculus, Volume I_ One-Variable Calculus, with an Introduction to Linear Algebra
Tom M. Apostol Calculus, Volume II_ Multi-Variable Calculus and Linear Algebra, with Applications to Differential Equations and Probability
Numerical Analysis - Richard L. Burden (Author), J. Douglas Faires (Author), Annette M. Burden (Author)
Numerical recipes in C -The Art of Scientific Computing
Computer oriented numerical methods - N Datta
Computer Fundamentals and Programming in C - J.B. Dixit

An analytical solution to a problem is one that has a "proof": a series of logical steps that can be followed and verified as correct. If you use the quadratic formula to solve for x in a quadratic equation, this is an analytical solution to the problem.

In contrast to this, some problems are solved via other means. If, instead of using the quadratic formula, you try a lot of values for x and y, this is a numerical solution. Some problems simply do not have analytical solutions and must be approximated using numerical methods -- for example, many complicated integrals.

for example ...

this is an instantaneous rate of change , find the function ...

and the answer is a function that has the property of ,this instantaneous rate of change ...


example 2 :

this is an instantaneous rate of change ...

find the function ,the solution , the answer which is a set of points that has certain underlying qualitative property which produced, this instantaneous rate of change ...

Say we were to solve the initial value problem:

y' = 2x

y(0) = 0

It's so simple, you could find a formulaic solution in your head, namely y = x2. On the other hand, say we were to use a numerical technique. (Yes, I know we don't know how to do this yet, but go with me on this for a second!) The resulting numerical solution would simply be a table of values. To get a better feel for the nature of these two types of solution, let's compare them side by side, along with the graphs we would get based on what we know about each one:

Notice that the graph derived from the formulaic solution is smoothly continuous, consisting of an infinite number of points on the interval shown. On the other hand, the graph based on the numerical solution consists of just a bare eight points, since the numerical method used apparently only found the value of the solution for x-increments of size 0.2.


Using Numerical Solutions

So what good is the numerical solution if it leaves out so much of the real answer? Well, we can respond to that question in several ways:

The numerical solution still looks like it is capturing the general trend of the "real" solution, as we can see when we look at the side-by-side graphs. This means that if we are seeking a qualitative view of the solution, we can still get it from the numerical solution, to some extent.

The numerical solution could even be "improved" by playing "join-the-dots" with the set of points it produces. In fact this is exactly what some solver packages, such as Mathematica, do do with these solutions. (Mathematica produces a join-the-dots function that it calls InterpolatingFunction.)

When actually using the solutions to differential equations, we often aren't so much concerned about the nature of the solution at all possible points. Think about it! Even when we are able to get formulaic solutions, a typical use we make of the formula is to substitute values of the independent variable into the formula in order to find the values of the solution at specific points. Did you hear that? Let me say it again: to find the values of the solution at specific
points. This is exactly what we can still do with a numerical solution
​

Solving an ordinary differential equation (ODE) or initial value problem means finding a clear expression for y in terms of a finite number of elementary functions of x. Euler’s method is one of the simplest method for the numerical solution of such equation or problem. This C program for Euler’s method considers an ordinary differential equations, and the initial values of x and y are known.

Mathematically, here, the curve of solution is approximated by a sequence of short lines i.e. by the tangent line in each interval. (Derivation) Using these information, the value of the value of ‘y_n’ corresponding to the value of ‘x_n‘ is to determined by dividing the length (x_n – x) into n strips.
Therefore, strip width= (x_n – x)/n and x_n=x₀+ nh. Again, if m be the slope of the curve at point, y₁₌ y_{0 + m}(x₀ , y_o)h.
Similarly, values of all the intermediate y can be found out.

Below is a source code forEuler’s method in C to solve the ordinary differential equation dy/dx = x+y. It asks for the value of of x_{0 ,} y_{0 ,}x_n and h. The value of slope at different points is calculated using the function ‘fun’.
The values of y are calculated in while loop which runs till the initial value of x is not equal to the final value. All the values of ‘y’ at corresponding ‘x’ are shown in the output screen. dy/dx = x+y

#include<stdio.h>
#include<math.h>
  
  
main()
{
                            
      float x;                                                                  /*defining variables*/
      float y;
      float h;
      float targetx;
        
       
        
      puts("This program will solve the differential equation y' = y - x \nusing Euler's Method with y(0)=1/2 \n\n");
      puts("Please enter the desired constant step size. (h-value)\n\n");
      scanf("%f", &h);                                                           /* Defining step size*/
      puts("\n\nNow enter the desired x-value to solve for y.\n\n");
      scanf("%f", &targetx);
        
      y = 0.5;
      x = 0.0;
        
      puts("\n\nX                Y");
        
      while ( x != targetx )
      {
        
      printf("\n\n%f     %f", x, y);
        
      y = y + ((y - x)*h);
        
      x= x+h;
      }
         
      printf("\n\n%f     %f\n", x, y);
        
      printf("\nThe value of y at the given x is %f.\n\n", y, h);
             
      system("pause");
        
}

here the solution is a set of points or a table of values that has certain underlying qualitative property which produced, this instantaneous rate of change ...

??